The moduli of smooth hypersurfaces with level structure

We construct the moduli space of smooth hypersurfaces with level N structure over Z[1/N] . As an application we show that, for N large enough, the stack of smooth hypersurfaces over Z[1/N] is uniformisable by a smooth affine scheme. To prove our results, we use the Lefschetz trace formula to show that automorphisms of smooth hypersurfaces act faithfully on their cohomology. We also prove a global Torelli theorem for smooth cubic threefolds over fields of odd characteristic.